Scalar curvature of minimal hypersurfaces in a sphere

We improve the well-known scalar curvature pinching theorem due to Peng–Terng for n ( n ⩽ 5 )-dimensional minimal hypersurfaces to the case of arbitrary n. Precisely, if M is a closed and minimal hypersurface in a unit sphere S n + 1 , then there exists a positive constant δ ( n ) depending only on n such that if n ⩽ S ⩽ n + δ ( n ) , then S ≡ n , i.e., M is a Clifford torus S k ( k n ) × S n − k ( n − k n ) , k = 1 , 2 , … , n − 1 .